Feb 3,8

Representations of tori. Weights. Weight multiplicity diagrams.
The Lie algebras of GL(n), SL(n), U(n), SU(n).
Weyl's unitary trick for those groups. Corollary: complete reducibility.
Thm: the irreps of SL(2) are exactly the Sym^k(C^2).

Dominance order on Z^n.

(Nonstandard) Definition: a representation V of U(n) is pointed at lambda if lambda is a weight of V, its multiplicity is 1, and lambda dominates all the other weights of V.

Feb 1

Orthonormality of characters extends to measurable representations of groups for which one can define an invariant, finite, measure, e.g. S^1.
We know enough irreps of S^1, namely z |-> z^n for each integer n, that (by Fourier theory) we must have found all the characters of irreps. So that's them and they're all 1-dim.

Def. Algebraic subgroup of GL(n).
Def. Algebraic representation, meaning all the matrix coefficients are polynomials in the original matrix entries or det^{-1}.
(Changes of basis are given by linear polynomials, so the definition doesn't really depend on choice of basis.)

Non-ex. C acts on C by z |-> exp(lambda z). That's a smooth rep, but not algebraic.
Ex. SL(2) acting on Sym^2 C^2. The matrix entries are homogeneous quadratic.

Thm. Let N_+ be the upper triangular 2x2 matrices with 1s on the diagonal.
Then in any continuous representation, there are 1-dim invariant subspaces.
If the representation is algebraic, then those subspaces are trivial representations.

Let's look for representations of G = SL(2). They should all occur inside algebraic functions on SL(2), which is C[a,b,c,d] / (det = 1).
Rather, for each irrep V, we should find V @ V^*.
So take the 1 x N_+ invariants, to try to cut down V^* to something smaller, but by the theorem, nonzero.
Claim. The only 1 x N_+ invariant functions on SL(2) are C[a,c].
(We'll complete this proof next time.)

Cor. Each irrep occurs inside some Sym^k(C^2).
Next time: the Sym^k(C^2) are actually irreducible, and each has a unique N_+-invt vector.

Jan 27

We proved:
Irreps of GxH are each the tensor product of an irrep of G with one of H.
C[G] is the sum over irreps V of V^* @ V, as a GxG-representation.
Character tables are square, i.e. the number of irreps is the number of conjugacy classes.
If G > H, two finite groups, then H misses some conjugacy class of G.
If G = U(n), H = T^n, then H hits every conjugacy class of G.

We didn't prove, but it's true:
As an algebra, C[G] is the direct sum of matrix algebras End(V), V running over irreps.

Jan 25

Overview:
1. Representation theory of finite groups.
2. ... of SL(2).
3. ... of U(n) and GL(n).
4. The adjoint representation of a Lie group; root system and Weyl group.
5. Classification of nice Lie groups.
6. Rep theory of general Lie groups.

Defs. Reps of finite groups on finite-dim complex vector spaces.
Irreducible, indecomposable.
Hom(V,W) as a rep. Equivariant maps.
Schur's lemma.
\pi_G|_V = 1/|G| \sum_G g|_V is a projection whose image is the G-invariants,
and whose trace is the dimension of the G-invariants.

Thm. Orthonormality of characters.
Cor. # irreps is at most # conjugacy classes.
Ex. S_3, S_4.

Welcome to Math 6500

The class homepage is here.

Final exam with answers

Here.

Answers to practice problems

1a. Let a,b be natural numbers, a < b. Assume there exist x,y in the naturals such that xa + yb = gcd(a,b). Show y < 10.

A. It depends on whether a=0 or a>0.

If a=0, then the gcd is b. x doesn't matter, and y must be 1. 1 < 10, huzzah.

If a>0, then the largest the gcd can be is a. On the other hand, the smallest positive number that xa+yb can be is a. So both must be a. Hence x=1, y=0. 0 < 10, huzzah redux.

b. Give an example where x > 10.

A. Apparently we're in the a=0 case, since otherwise x would have to be 1.
So y must be 1, and x and b are free to roam. Take x = 57 and b = 1066, for example.

2. What are the last two decimal digits of 11²⁰⁰⁸?

A. 81.
More generally, 11ⁿ == 10n + 1 mod 100.
This is easy to prove by induction: multiply both sides by 11, obtaining
11ⁿ⁺¹ == 110n + 11 == 10n + 11 == 10(n+1) + 1 mod 100.

3a. Draw an example of finite sets X,Y,Z and functions f:X->Y, g:Y->Z where |X| > |Y|, f is not onto, but the composite g o f is onto.

A. Say X = {a,b,c}, Y = {taupe,vermilion}, Z = {cheese}. Let f(x) = taupe for any x, and g be the only function Y->Z.

b. Prove that |Y| is not equal to |Z| (not just in your example, but in any example).

A. Since g o f is onto, the map image(f) -> Z (made by restricting g) is also onto.
Hence |Z| is less than or equal to |image(f)|. Since image(f) is contained in Y, |image(f)| is less than or equal to |Y|. But f is not onto, so it's strictly contained. Hence |image(f)| < |Y|.


4. In each of the following: can you turn the following two congruences into one, and if so, what?

v1. b congruent to 2 mod 6, b congruent to 7 mod 10.

A. No you can't; the first says b is even, the second says it's odd.
Whereas any single congruence does have solutions.

v2. b congruent to 1 mod 6, b congruent to 7 mod 10.

A. The first can be rewritten as "b congruent to 7 mod 6". Hence 6 and 10 divide b-7, or equivalently, 30 divides b-7. So they can be rewritten as e.g. b-2 congruent to 5 mod 30.

5. Say b == a+7 mod 27, and a² == b² mod 27.
Figure out a,b mod 27.

A. a² - b² == 0 mod 27.
So (a+b)(a-b) == 0 mod 27.
So (a+b)(-7) == 0 mod 27.
Since gcd(7,27) = 1, we can cancel the 7.
So a+b == 0 mod 27.
So 2a+7 == 0 mod 27.
How to cancel the 2? Subtract 27 from both sides, for example; 2a-20 == 27 == 0 mod 27.
So a-10 == 0 mod 27. Hence a == 10 mod 27, b == 17 == -a mod 27.

109 roundup

Definitions.
Quantifiers.
Proofs. Case check, contradiction, induction, series of reductions.
Make sure during induction that the chain of implications isn't broken anywhere.
To prove sets are equal, show both inclusions.
To prove functions are equal, check them on every element.
Functions; 1:1, onto, invertible.
Baby number theory. The division algorithm, Euclid's algorithm, the fact that the gcd can be written as an integer linear combination of the two numbers, unique factorization into primes. Congruences. Chinese Remainder Theorem.
Partitions and equivalence relations.
(n choose k) and the Pascal recurrence for it.

Office hours

I'll be in my office (7450 APM) Tuesday 11-4 and Wednesday 11-2. I'll be giving preference to my Math 109 students on Tuesday and my Ma 20b students on Wednesday, but anyone will be welcome any time.
If you want to call ahead to check "Are you already tied up with the other class?", feel free; my office number is 858-534-6450.
(Don't get 7450 and 6450 confused!)